Operator-norm convergence estimates for elliptic homogenisation problems on periodic singular structures

TL;DR

This paper establishes order O(ε) operator-norm resolvent estimates for elliptic problems with periodic coefficients on arbitrary periodic measures, including singular structures supported on lower-dimensional manifolds.

Contribution

It provides the first order O(ε) operator-norm resolvent estimates for elliptic problems on general periodic measures, extending results to singular multistructures.

Findings

Order O(ε) resolvent estimates proven

Applicable to both absolutely continuous and singular measures

Includes multistructures supported on lower-dimensional manifolds

Abstract

For a an arbitrary periodic Borel measure $\mu$, we prove order $O(\varepsilon)$ operator-norm resolvent estimates for the solutions to scalar elliptic problems in $L^2({\mathbb R}^d, d\mu^\varepsilon)$ with $\varepsilon$-periodic coefficients, $\varepsilon>0.$ Here $\mu^\varepsilon$ is the measure obtained by $\varepsilon$-scaling of $\mu.$ Our analysis includes both the case of a measure absolutely continuous with respect to the standard Lebesgue measure and the case of "singular" periodic structures (or "multistructures"), when $\mu$ is supported by lower-dimensional manifolds.